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Statistical Approaches to Learning and Discovery
Graphical Models
Zoubin Ghahramani & Teddy Seidenfeld
[email protected] & [email protected]
CALD / CS / Statistics / Philosophy
Carnegie Mellon University
Spring 2002
Some Examples
XK
X1
Λ
Y1
Y2
factor analysis
probabilistic PCA
ICA
YD
X1
X2
X3
XT
Y1
Y2
Y3
YT
hidden Markov models
linear dynamical systems
U1
U2
U3
S1
S2
S3
X1
X2
X3
Y1
Y2
Y3
switching state-space models
Markov Networks (Undirected Graphical Models)
B
A
Examples:
Boltzmann Machines
Markov Random Fields
C
D
E
Semantics: Every node is conditionally independent from its non-neighbors given
its neighbors.
Conditional Independence: X⊥
⊥Y |V ⇔ p(X|Y, V ) = p(X|V ) when p(Y, V ) > 0.
also X⊥
⊥Y |V ⇔ p(X, Y |V ) = p(X|V )p(Y |V ).
Markov Blanket: V is a Markov Blanket for X iff X⊥
⊥Y |V for all Y ∈
/ V.
Markov Boundary: minimal Markov Blanket
Clique Potentials and Markov Networks
Definition: a clique is a fully connected subgraph (usually maximal).
Ci will denote the set of variables in the ith clique.
B
A
C
D
E
1. Identify cliques of graph G
2. For each clique Ci assign a non-negative function gi(Ci) which measures
“compatibility”.
Q
P
Q
3. p(X1, . . . , Xn) = Z1 i gi(Ci) where Z = X1···Xn i gi(Ci) is the normalization
The graph G embodies the conditional independencies in p (i.e. G is a Markov Field
relative to p):
If V lies in all paths between X and Y in G, then X⊥
⊥Y |V .
Hammersley–Clifford Theorem (1971)
Theorem: A probability function p formed by a normalized product of positive
functions on cliques of G is a Markov Field relative to G.
Definition: The graph G is a Markov Field relative to p if it does not imply any
conditional independence relationships that are not true in p.
(We are usually interested in the minimal such graph.)
Proof: We need to show that the neighbors of X, ne(X) are a Markov Blanket
for X:
Y
1Y
1 Y
p(X, Y, . . .) =
gi(Ci) =
gi(Ci)
gj (Cj )
Z i
Z
i:X∈Ci
j:X ∈C
/ j
1
1
= f1 X, ne(X) f2 ne(X), Y = 0 p(X| ne(X)) p(Y | ne(X))
Z
Z
This shows that:
p(X, Y | ne(X)) = p(X| ne(X)) p(Y | ne(X)) ⇔ X⊥
⊥Y | ne(X).
Problems with Markov Networks
Many useful independencies are unrepresented — two variables are connected merely
because some other variable depends on them:
Rain
Sprinkler
Rain
Ground wet
Marginal independence vs. conditional independence.
“Explaining Away”
Sprinkler
Ground wet
Bayesian Networks (Directed Graphical Models)
B
A
C
D
E
Semantics: X⊥
⊥Y |V if V d-separates X from Y .
Definition: V d-separates X from Y if along every undirected path between X and
Y there is a node W such that either:
1. W has converging arrows along the path (→ W ←) and neither W nor its
descendants are in V , or
2. W does not have converging arrows along the path (→ W →) and W ∈ V .
The “Bayes-ball” algorithm.
Corollary:
Markov Blanket
parents-of-children(X)}.
for
X:
{parents(X) ∪ children(X) ∪
Bayesian Networks (Directed Graphical Models)
B
A
C
D
E
A Bayesian network corresponds to a factorization of the joint probability distribution:
p(A, B, C, D, E) = p(A)p(B)p(C|A, B)p(D|B, C)p(E|C, D)
In general:
p(X1, . . . , Xn) =
n
Y
i=1
where pa(i) are the parents of node i.
p(Xi|Xpa(i))
From Bayesian Trees to Markov Trees
1
5
3
2
4
6
7
p(1, 2, . . . , 7) = p(3)p(1|3)p(2|3)p(4|3)p(5|4)p(6|4)p(7|4)
p(1, 3)p(2, 3)p(3, 4)p(4, 5)p(4, 6)p(4, 7)
=
p(3)p(3)p(4)p(4)p(4)
product of cliques
=
product of clique intersections
= g(1, 3)g(2, 3)g(3, 4)g(4, 5)g(4, 6)g(4, 7) =
Y
i
gi(Ci)
Belief Propagation (in Singly Connected Bayesian Networks)
Definition: S.C.B.N. has an undirected underlying graph which is a tree, ie there is
only one path between any two nodes.
Goal: For some node X we want to compute p(X|e) given evidence e.
Since we are considering S.C.B.N.s:
−
• every node X divides the evidence into upstream e+
and
downstream
e
X
X
+
• every arc X → Y divides the evidence into upstream eXY and downstream e−
XY .
The three key ideas behind Belief Propagation
Idea 1: Our belief about the variable X can be found by combining upstream and
downstream evidence:
−
p(X, e+
,
e
p(X, e)
+
X X)
p(X|e) =
=
∝
p(X|e
X)
+ −
p(e)
p(eX , eX )
p(e−
|X, e+
)
X
|
{z X}
+
X d-separates e−
X from eX
×
−
= p(X|e+
)p(e
X
X |X) = π(X)λ(X)
Idea 2: The upstream and downstream evidence can be computed via a local
message passing algorithm between the nodes in the graph.
Idea 3: “Don’t send back to a node (any part of) the message it sent to you!”
Belief Propagation
U1
......
top-down causal support:
πX (Ui) = p(Ui|e+
Ui X )
Un
X
bottom-up diagnostic support:
λYj (X) = p(e−
XYj |X)
Y1
......
Ym
To update the belief about X:
1
BEL(X) = λ(X)π(X)
Z
Y
λ(X) =
λYj (X)
j
π(X) =
X
U1 ···Un
p(X|U1, . . . , Un)
Y
i
πX (Ui)
Belief Propagation, cont
U1
......
top-down causal support:
πX (Ui) = p(Ui|e+
Ui X )
Un
X
bottom-up diagnostic support:
λYj (X) = p(e−
XYj |X)
Y1
......
Ym
Bottom-up propagation, X sends to Ui:
X
Y
1X
λX (Ui) =
λ(X)
p(X|U1, . . . , Un)
πX (Uk )
Z
X
Uk :k6=i
k6=i
Top-down propagation, X sends to Yj :
Y
X
1 BEL(X)
1Y
πYj (X) =
λYk (X)
p(X|U1, . . . , Un)
πX (Ui) =
Z
Z λYj (X)
i
k6=j
U1 ···Un
Belief Propagation in multiply connected Bayesian Networks
The Junction Tree algorithm: Form an undirected graph from your directed graph
such that no additional conditional independence relationships have been created
(this step is called “moralization”). Lump variables in cliques together and form a
tree of cliques—this may require a nasty step called “triangulation”. Do inference
in this tree.
Cutset Conditioning: or “reasoning by assumptions”. Find a small set of variables
which, if they were given (i.e. known) would render the remaining graph singly
connected. For each value of these variables run belief propagation on the singly
connected network. Average the resulting beliefs with the appropriate weights.
Loopy Belief Propagation: just use BP although there are loops. In this case
the terms “upstream” and “downstream” are not clearly defined. No guarantee of
convergence, but often works well in practice.
Learning with Hidden Variables:
The EM Algorithm
θ1
X1
θ3
θ2
X3
X2
θ4
Y
Assume a model parameterised by θ with observable variables Y and hidden variables
X
Goal: maximise log likelihood of observables.
L(θ) = ln p(Y |θ) = ln
X
p(Y, X|θ)
X
• E-step: first infer p(X|Y, θold), then
• M-step: find θnew using complete data learning
The E-step requires solving the inference problem: finding explanations, X,
for the data, Y , given the current model, θ (using e.g. BP).
Expressive Power of Bayesian and Markov Networks
No Bayesian network can
represent these and only these
independencies
No matter how we direct the arrows there will always be two non-adjacent parents
sharing a common child =⇒ dependence in Bayesian network but independence in
Markov network.
No
Markov
network
can
represent these and only these
independencies